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minhthien_2016
minhthien_2016
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How can I find the integer numbers a, b, c, m,n so that the funtion f[x]=(a x^2 + b x + c)/(m x + n) with the equation D[f[x], x]] == 0 have two integer solutions x1, x2 and the values f[x1] and f[x2] are also integer numbers?

I tried

Clear["Global`*"]
f[x_] = (a x^2 + b x + c)/(m x + n);
x1 = (-a n - Sqrt[a c m^2 - a b m n + a^2 n^2])/(a m);
x2 = (-a n + Sqrt[a c m^2 - a b m n + a^2 n^2])/(a m);
list = Table[
  If[GCD[a, b, c] == 1 && GCD[m, n] == 1 && 
    a c m^2 - a b m n + a^2 n^2 > 0 && a n - b m  != 0 && 
    a (c m^2 - b m n + a n^2) > 0 && b^2 - 4 * a * c < 0 && 
    IntegerQ[x1] && IntegerQ[x2] && b c m n x1 x2 f[x1]* f[x2] != 0 &&
     IntegerQ[f[x1]] && 
    IntegerQ[f[x2]], {(a x^2 + b x  + c)/(m x + n), {x1, f[x1]}, {x2, 
     f[x2]}}, Nothing], {a, 1, 10}, {b, 1, 10}, {c, 1, 10}, {m, 1, 10}, {n, 1, 10}]

enter image description here

How can I remove the empty setsets in my results?

How can I find the integer numbers a, b, c, m,n so that the funtion f[x]=(a x^2 + b x + c)/(m x + n) with the equation D[f[x], x]] == 0 have two integer solutions x1, x2 and the values f[x1] and f[x2] are also integer numbers?

I tried

Clear["Global`*"]
f[x_] = (a x^2 + b x + c)/(m x + n);
x1 = (-a n - Sqrt[a c m^2 - a b m n + a^2 n^2])/(a m);
x2 = (-a n + Sqrt[a c m^2 - a b m n + a^2 n^2])/(a m);
list = Table[
  If[GCD[a, b, c] == 1 && GCD[m, n] == 1 && 
    a c m^2 - a b m n + a^2 n^2 > 0 && a n - b m  != 0 && 
    a (c m^2 - b m n + a n^2) > 0 && b^2 - 4 * a * c < 0 && 
    IntegerQ[x1] && IntegerQ[x2] && b c m n x1 x2 f[x1]* f[x2] != 0 &&
     IntegerQ[f[x1]] && 
    IntegerQ[f[x2]], {(a x^2 + b x  + c)/(m x + n), {x1, f[x1]}, {x2, 
     f[x2]}}, Nothing], {a, 1, 10}, {b, 1, 10}, {c, 1, 10}, {m, 1, 10}, {n, 1, 10}]

enter image description here

How can I remove the empty set in my results?

How can I find the integer numbers a, b, c, m,n so that the funtion f[x]=(a x^2 + b x + c)/(m x + n) with the equation D[f[x], x]] == 0 have two integer solutions x1, x2 and the values f[x1] and f[x2] are also integer numbers?

I tried

Clear["Global`*"]
f[x_] = (a x^2 + b x + c)/(m x + n);
x1 = (-a n - Sqrt[a c m^2 - a b m n + a^2 n^2])/(a m);
x2 = (-a n + Sqrt[a c m^2 - a b m n + a^2 n^2])/(a m);
list = Table[
  If[GCD[a, b, c] == 1 && GCD[m, n] == 1 && 
    a c m^2 - a b m n + a^2 n^2 > 0 && a n - b m  != 0 && 
    a (c m^2 - b m n + a n^2) > 0 && b^2 - 4 * a * c < 0 && 
    IntegerQ[x1] && IntegerQ[x2] && b c m n x1 x2 f[x1]* f[x2] != 0 &&
     IntegerQ[f[x1]] && 
    IntegerQ[f[x2]], {(a x^2 + b x  + c)/(m x + n), {x1, f[x1]}, {x2, 
     f[x2]}}, Nothing], {a, 1, 10}, {b, 1, 10}, {c, 1, 10}, {m, 1, 10}, {n, 1, 10}]

enter image description here

How can I remove the empty sets in my results?

deleted 222 characters in body
Source Link
John Paul Peter
John Paul Peter
  • 1.8k
  • 4
  • 11

How to find the integer numbers a, b, c, m, n satifying the following conditions and remove the empty sets in this results?

Clear["Global`*"]
f[x_] = (a x^2 + b x + c)/(m x + n);
Solve[f'[x] ==x1 0,= x];
FullSimplify[f[(-a n - Sqrt[a c m^2 - a b m n + a^2 n^2])/(a m)]];;
FullSimplify[f[x2 = (-a n + Sqrt[a c m^2 - a b m n + a^2 n^2])/(a m)]];

Table[If[ ;
  alist c= m^2Table[
 - aIf[GCD[a, b, mc] n== +1 a^2&& n^2GCD[m, >=n] 0== 1 && 
   IntegerQ[(-a n - Sqrt[aa c m^2 - a b m n + a^2 n^2])/(a m)] &&n^2 
 > 0 && IntegerQ[(-a n + Sqrt[a c m^2 - a b m n + a^2 n^2])/(a!= m)]0 && 
   IntegerQ[(b m - 2 (a n + Sqrt[a (c m^2 - b m n + a n^2)]))/m^2] && 
 > 0 IntegerQ[(b&& mb^2 - 2 a n4 +* 2a Sqrt[a* (c m^2 - b< m0 n&& + 
 a n^2)])/
  IntegerQ[x1] && m^2],IntegerQ[x2] {{a,&& b, c, m, 
    n}, {(-a n - Sqrt[a c m^2 - a b mx1 nx2 +f[x1]* a^2f[x2] n^2])/(a!= m),0 &&
    f[(-a n - Sqrt[a c m^2 - aIntegerQ[f[x1]] b&& m 
 n + a^2 n^2])/(a m)]}IntegerQ[f[x2]], {(-a nx^2 + 
     Sqrt[a c m^2 - a b mx n + a^2 n^2]c)/(a m), 
    f[(-a nx + Sqrt[a c m^2n), -{x1, af[x1]}, b{x2, m 
 n + a^2 n^2])/(a m)]f[x2]}}, 
  Nothing], {a, 1, 10}, {b, 1, 10}, {c, 1, 10}, {m, 1, 10}, {n, 1, 10}]

Some output empty or Indeterminate.enter image description here

How can I get correct resultremove the empty set in my results?

enter image description here

How to find the integer numbers a, b, c, m, n satifying the following conditions?

f[x_] = (a x^2 + b x + c)/(m x + n);
Solve[f'[x] == 0, x];
FullSimplify[f[(-a n - Sqrt[a c m^2 - a b m n + a^2 n^2])/(a m)]];
FullSimplify[f[(-a n + Sqrt[a c m^2 - a b m n + a^2 n^2])/(a m)]];

Table[If[ 
  a c m^2 - a b m n + a^2 n^2 >= 0 && 
   IntegerQ[(-a n - Sqrt[a c m^2 - a b m n + a^2 n^2])/(a m)] && 
    IntegerQ[(-a n + Sqrt[a c m^2 - a b m n + a^2 n^2])/(a m)] && 
   IntegerQ[(b m - 2 (a n + Sqrt[a (c m^2 - b m n + a n^2)]))/m^2] && 
   IntegerQ[(b m - 2 a n + 2 Sqrt[a (c m^2 - b m n + a n^2)])/
    m^2], {{a, b, c, m, 
    n}, {(-a n - Sqrt[a c m^2 - a b m n + a^2 n^2])/(a m), 
    f[(-a n - Sqrt[a c m^2 - a b m n + a^2 n^2])/(a m)]}, {(-a n + 
     Sqrt[a c m^2 - a b m n + a^2 n^2])/(a m), 
    f[(-a n + Sqrt[a c m^2 - a b m n + a^2 n^2])/(a m)]}}, 
  Nothing], {a, 1, 10}, {b, 1, 10}, {c, 1, 10}, {m, 1, 10}, {n, 1, 10}]

Some output empty or Indeterminate. How can I get correct result?

enter image description here

How to find the integer numbers a, b, c, m, n satifying the following conditions and remove the empty sets in this results?

Clear["Global`*"]
f[x_] = (a x^2 + b x + c)/(m x + n);
x1 = (-a n - Sqrt[a c m^2 - a b m n + a^2 n^2])/(a m);
x2 = (-a n + Sqrt[a c m^2 - a b m n + a^2 n^2])/(a m);
list = Table[
  If[GCD[a, b, c] == 1 && GCD[m, n] == 1 && 
    a c m^2 - a b m n + a^2 n^2 > 0 && a n - b m  != 0 && 
    a (c m^2 - b m n + a n^2) > 0 && b^2 - 4 * a * c < 0 &&  
    IntegerQ[x1] && IntegerQ[x2] && b c m n x1 x2 f[x1]* f[x2] != 0 &&
     IntegerQ[f[x1]] &&  
    IntegerQ[f[x2]], {(a x^2 + b x  + c)/(m x + n), {x1, f[x1]}, {x2,  
     f[x2]}}, Nothing], {a, 1, 10}, {b, 1, 10}, {c, 1, 10}, {m, 1, 10}, {n, 1, 10}]

enter image description here

How can I remove the empty set in my results?

added 782 characters in body; edited tags; edited title
Source Link
John Paul Peter
John Paul Peter
  • 1.8k
  • 4
  • 11

How to find the integer numbers a, b, c, d, e, m, n satifying the following conditions?

How can I know that,find the functioninteger numbers f[x_]a, :=b, c, m,n so that the funtion f[x]=(a x^2 + 6b x + 5c)/(x^2 + 2m x + 3n) with the equation D[f[x], x]x]] == 0 have two integer solutions -2, 1x1, x2 and the values f[-2]=1f[x1], and f[1]=2f[x2]. My code are also integer numbers?

I tried

f[x_] := (a x^2 + 6b x + 5c)/(x^2m x + n);
Solve[f'[x] == 0, x];
FullSimplify[f[(-a n - Sqrt[a c m^2 - a b m n + a^2 n^2])/(a m)]];
FullSimplify[f[(-a n + Sqrt[a c m^2 - a b m n + a^2 n^2])/(a m)]];

Table[If[ 
  a c m^2 - a b m n + a^2 n^2 >= 0 && 
   IntegerQ[(-a n - Sqrt[a c m^2 - a b m n + a^2 n^2])/(a m)] && 
   IntegerQ[(-a n + Sqrt[a c m^2 - a b m n + a^2 n^2])/(a m)] && 
   IntegerQ[(b m - 2 x(a n + 3Sqrt[a (c m^2 - b m n + a n^2)]))/m^2] && 
Reduce[Simplify[D[f[x]   IntegerQ[(b m - 2 a n + 2 Sqrt[a (c m^2 - b m n + a n^2)])/
    m^2], x]]{{a, ==b, 0c, x]m, 
    n}, {(-a n - Sqrt[a c m^2 - a b m n + a^2 n^2])/(a m), 
    f[(-2]a n - Sqrt[a c m^2 - a b m n + a^2 n^2])/(a m)]}, {(-a n + 
f[1]     Sqrt[a c m^2 - a b m n + a^2 n^2])/(a m), 
    f[(-a n + Sqrt[a c m^2 - a b m n + a^2 n^2])/(a m)]}}, 
  Nothing], {a, 1, 10}, {b, 1, 10}, {c, 1, 10}, {m, 1, 10}, {n, 1, 10}]

Some output empty or Indeterminate. How can I find the integer numbers a, b, c, d, e, m, where 0<a, b, c, d, e, m <=1 so that the funtion f[x]=(a x^2 + b x + c)/(d x^2 + e x + m) with the equation D[f[x], x]] == 0 have two integer solutions x1, x2 and the values f[x1] and f[x2] are also integer numbersget correct result?

enter image description here

How to find the integer numbers a, b, c, d, e, m satifying the following conditions?

I know that, the function f[x_] := (x^2 + 6 x + 5)/(x^2 + 2 x + 3) with D[f[x], x] have two integer solutions -2, 1 and f[-2]=1, f[1]=2. My code

f[x_] := (x^2 + 6 x + 5)/(x^2 + 2 x + 3)
Reduce[Simplify[D[f[x], x]] == 0, x]
f[-2]
f[1]

How can I find the integer numbers a, b, c, d, e, m, where 0<a, b, c, d, e, m <=1 so that the funtion f[x]=(a x^2 + b x + c)/(d x^2 + e x + m) with the equation D[f[x], x]] == 0 have two integer solutions x1, x2 and the values f[x1] and f[x2] are also integer numbers?

How to find the integer numbers a, b, c, m, n satifying the following conditions?

How can I find the integer numbers a, b, c, m,n so that the funtion f[x]=(a x^2 + b x + c)/(m x + n) with the equation D[f[x], x]] == 0 have two integer solutions x1, x2 and the values f[x1] and f[x2] are also integer numbers?

I tried

f[x_] = (a x^2 + b x + c)/(m x + n);
Solve[f'[x] == 0, x];
FullSimplify[f[(-a n - Sqrt[a c m^2 - a b m n + a^2 n^2])/(a m)]];
FullSimplify[f[(-a n + Sqrt[a c m^2 - a b m n + a^2 n^2])/(a m)]];

Table[If[ 
  a c m^2 - a b m n + a^2 n^2 >= 0 && 
   IntegerQ[(-a n - Sqrt[a c m^2 - a b m n + a^2 n^2])/(a m)] && 
   IntegerQ[(-a n + Sqrt[a c m^2 - a b m n + a^2 n^2])/(a m)] && 
   IntegerQ[(b m - 2 (a n + Sqrt[a (c m^2 - b m n + a n^2)]))/m^2] && 
   IntegerQ[(b m - 2 a n + 2 Sqrt[a (c m^2 - b m n + a n^2)])/
    m^2], {{a, b, c, m, 
    n}, {(-a n - Sqrt[a c m^2 - a b m n + a^2 n^2])/(a m), 
    f[(-a n - Sqrt[a c m^2 - a b m n + a^2 n^2])/(a m)]}, {(-a n + 
     Sqrt[a c m^2 - a b m n + a^2 n^2])/(a m), 
    f[(-a n + Sqrt[a c m^2 - a b m n + a^2 n^2])/(a m)]}}, 
  Nothing], {a, 1, 10}, {b, 1, 10}, {c, 1, 10}, {m, 1, 10}, {n, 1, 10}]

Some output empty or Indeterminate. How can I get correct result?

enter image description here

added 36 characters in body
Source Link
John Paul Peter
John Paul Peter
  • 1.8k
  • 4
  • 11
Source Link
John Paul Peter
John Paul Peter
  • 1.8k
  • 4
  • 11